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Bifurcation and complexity of Monod type predator–prey system in a pulsed chemostat. (English) Zbl 1096.34029
The authors introduce and investigate a predator-prey model that involves a periodically pulsed substrate as a third unknown variable. The right-hand side of this three-dimensional system consists of linear terms and two kinds of nonlinear terms expressing a satiation effect of prey and predator if the substrate resp. prey concentrations are sufficiently large. In the theoretical part of the paper, the existence of a positive periodic solution is proved whose stability depends on the parameters of the system, such that for certain parameter regions, only the zero solution for both predator and prey or simply the predator is globally asymptotically stable. Further results on possible bifurcation are obtained by numerical investigation. Here, two different parameters and, moreover, the period of the pulses act as bifurcation parameters. In this way, different routes to chaos are documented, e.g., a cascade of period doubling, strange attractor, chaotic region with period windows, and finally a cascade of period halving.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34C28 Complex behavior and chaotic systems of ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
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